Actually the phrase is not "American natives" but "Native Americans".
I think what the writer of that definition was trying to say was that the word "native" as a stand-alone noun to mean a person from a non-Western culture with a low level of technology is now considered offensive. Like if you drew a picture of a group of people standing in front of a mud hut, with painted faces and carrying spears, and labeled it "Natives", this would be considered offensive.
The word "native" in general simply means some one or thing that originally comes from a particular place. In this sense it is a perfectly good word. I certainly would not be offended if you referred to me as "a native of New York". We routinely talk about "foods native to the region", about a person's "native language", etc.
But anyway, I don't think there's any simple rule as to what makes a word or phrase offensive. When I was a boy in the 1960s, members of a certain ethnic group were routinely referred to as "negroes". Then about the 1970s or so we were told that this term was offensive, and that we should call them "black". Then in the 1990s we were told that "black" is offensive, and we should call them "African-American". How did "black" go from being the polite term to being offensive? It just did. There's no pattern to such things.
I saw a survey a few years back that found that a majority of American Indians prefer to be called Indians rather than Native Americans. For that matter, I saw a survey fairly recently where they asked black people what they prefer to be called, and 1% said "African Americans", 2% said "black", 96% said "don't care", and there was the usual scattering of other answers.